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  • 2022.1
  • 12/20/2021
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DGELS Example Program in Fortran

* Copyright (C) 2009-2015 Intel Corporation. All Rights Reserved. * The information and material ("Material") provided below is owned by Intel * Corporation or its suppliers or licensors, and title to such Material remains * with Intel Corporation or its suppliers or licensors. The Material contains * proprietary information of Intel or its suppliers and licensors. The Material * is protected by worldwide copyright laws and treaty provisions. No part of * the Material may be copied, reproduced, published, uploaded, posted, * transmitted, or distributed in any way without Intel's prior express written * permission. No license under any patent, copyright or other intellectual * property rights in the Material is granted to or conferred upon you, either * expressly, by implication, inducement, estoppel or otherwise. Any license * under such intellectual property rights must be express and approved by Intel * in writing. * ============================================================================= * * DGELS Example. * ============== * * Program computes the least squares solution to the overdetermined linear * system A*X = B with full rank matrix A using QR factorization, * where A is the coefficient matrix: * * 1.44 -7.84 -4.39 4.53 * -9.96 -0.28 -3.24 3.83 * -7.55 3.24 6.27 -6.64 * 8.34 8.09 5.28 2.06 * 7.08 2.52 0.74 -2.47 * -5.45 -5.70 -1.19 4.70 * * and B is the right-hand side matrix: * * 8.58 9.35 * 8.26 -4.43 * 8.48 -0.70 * -5.28 -0.26 * 5.72 -7.36 * 8.93 -2.52 * * Description. * ============ * * The routine solves overdetermined or underdetermined real linear systems * involving an m-by-n matrix A, or its transpose, using a QR or LQ * factorization of A. It is assumed that A has full rank. * * Several right hand side vectors b and solution vectors x can be handled * in a single call; they are stored as the columns of the m-by-nrhs right * hand side matrix B and the n-by-nrhs solution matrix X. * * Example Program Results. * ======================== * * DGELS Example Program Results * * Solution * -0.45 0.25 * -0.85 -0.90 * 0.71 0.63 * 0.13 0.14 * * Residual sum of squares for the solution * 195.36 107.06 * * Details of QR factorization * -17.54 -4.76 -1.96 0.42 * -0.52 12.40 7.88 -5.84 * -0.40 -0.14 -5.75 4.11 * 0.44 -0.66 -0.20 -7.78 * 0.37 -0.26 -0.17 -0.15 * -0.29 0.46 0.41 0.24 * ============================================================================= * * .. Parameters .. INTEGER M, N, NRHS PARAMETER ( M = 6, N = 4, NRHS = 2 ) INTEGER LDA, LDB PARAMETER ( LDA = M, LDB = M ) INTEGER LWMAX PARAMETER ( LWMAX = 100 ) * * .. Local Scalars .. INTEGER INFO, LWORK * * .. Local Arrays .. DOUBLE PRECISION A( LDA, N ), B( LDB, NRHS ), WORK( LWMAX ) DATA A/ $ 1.44,-9.96,-7.55, 8.34, 7.08,-5.45, $ -7.84,-0.28, 3.24, 8.09, 2.52,-5.70, $ -4.39,-3.24, 6.27, 5.28, 0.74,-1.19, $ 4.53, 3.83,-6.64, 2.06,-2.47, 4.70 $ / DATA B/ $ 8.58, 8.26, 8.48,-5.28, 5.72, 8.93, $ 9.35,-4.43,-0.70,-0.26,-7.36,-2.52 $ / * * .. External Subroutines .. EXTERNAL DGELS EXTERNAL PRINT_MATRIX, PRINT_VECTOR_NORM * * .. Intrinsic Functions .. INTRINSIC INT, MIN * * .. Executable Statements .. WRITE(*,*)'DGELS Example Program Results' * * Query the optimal workspace. * LWORK = -1 CALL DGELS( 'No transpose', M, N, NRHS, A, LDA, B, LDB, WORK, $ LWORK, INFO ) LWORK = MIN( LWMAX, INT( WORK( 1 ) ) ) * * Solve the equations A*X = B. * CALL DGELS( 'No transpose', M, N, NRHS, A, LDA, B, LDB, WORK, $ LWORK, INFO ) * * Check for the full rank. * IF( INFO.GT.0 ) THEN WRITE(*,*)'The diagonal element ',INFO,' of the triangular ' WRITE(*,*)'factor of A is zero, so that A does not have full ' WRITE(*,*)'rank; the least squares solution could not be ' WRITE(*,*)'computed.' STOP END IF * * Print least squares solution. * CALL PRINT_MATRIX( 'Least squares solution', N, NRHS, B, LDB ) * * Print residual sum of squares for the solution * CALL PRINT_VECTOR_NORM( $ 'Residual sum of squares for the solution', M-N, NRHS, $ B( N+1, 1 ), LDB ) * * Print details of QR factorization. * CALL PRINT_MATRIX( 'Details of QR factorization', M, N, A, LDA ) STOP END * * End of DGELS Example. * * ============================================================================= * * Auxiliary routine: printing a matrix. * SUBROUTINE PRINT_MATRIX( DESC, M, N, A, LDA ) CHARACTER*(*) DESC INTEGER M, N, LDA DOUBLE PRECISION A( LDA, * ) * INTEGER I, J * WRITE(*,*) WRITE(*,*) DESC DO I = 1, M WRITE(*,9998) ( A( I, J ), J = 1, N ) END DO * 9998 FORMAT( 11(:,1X,F6.2) ) RETURN END * * Auxiliary routine: printing norms of matrix columns. * SUBROUTINE PRINT_VECTOR_NORM( DESC, M, N, A, LDA ) CHARACTER*(*) DESC INTEGER M, N, LDA DOUBLE PRECISION A( LDA, * ) * DOUBLE PRECISION TEMP INTEGER I, J * WRITE(*,*) WRITE(*,*) DESC DO J = 1, N TEMP = 0.0 DO I = 1, M TEMP = TEMP + A( I, J )*A( I, J ) END DO WRITE(*,9998,ADVANCE='NO') TEMP END DO WRITE(*,*) * 9998 FORMAT( 11(:,1X,F6.2) ) RETURN END

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